[8] Specimens of frosted or crackled Venetian glass are to be seen in the Slade Collection, British Museum, and fully justify Galileo’s comparison.

[9] Webb, Celestial Objects for Common Telescopes, p. 104, suggests this identification.

[10] In the list of the heights of lunar mountains determined by Beer and Maedler, given in their work Der Mond (Berlin, 1837), there are six which exceed 3000 toises, or 19,000 British feet.

[11] The illumination of the Moon in eclipses, noticed by Galileo, is now referred to the refraction of the sunlight by the earth’s atmosphere, and the reddish colour of the light is explained by Herschel (Outlines of Astronomy, ch. vii.) to be due to the absorption of the violet and blue rays by the aqueous vapour of the Earth’s atmosphere. The idea of a sensible lunar atmosphere is not in accordance with the observed phenomena of the occultations of stars.

[12] Galileo’s Systema Mundi. Owing to the violent opposition provoked by the discussion of the discoveries of Galileo, and their bearing on the Copernican system of astronomy, Galileo seems to have found it necessary to delay the publication of this work until 1632, when, believing himself safe under the friendship and patronage of Pope Urban VIII. and others in power at Rome, he at length published it. Urban, however, now turned against him, denounced the book and its author, and summoned him to Rome, where the well-known incidents of his trial and condemnation took place.

[13] The immense distance of stars makes it impossible for them to be magnified by any telescope, however powerful; the apparent or spurious disc is an optical effect, which depends on the telescope used, and is smallest with the largest aperture.

[14] The times of Galileo’s observations are to be understood as reckoned from sunset.

[15] The satellites of Jupiter revolve in planes very nearly, although not exactly, coincident with that of the equator of the planet, which is inclined 3° 5´ 30´´ to the orbit of the planet, and the plane of the orbit is inclined 1° 18´ 51´´ to the ecliptic.

[16] Galileo continues to call these bodies stars, perhaps meaning “Medicean stars,” throughout the description of their configurations, but as he had now detected their nature, it is more convenient to call them satellites, the term introduced by Kepler.

[17] In the edition of Galileo’s works published at Florence, 1854, there are given the tables of the hourly movements of the satellites of Jupiter, from which Galileo determined their periods of revolution. In the beginning of his treatise on floating bodies, Discorso intorno i Galleggianti, 1611-12, Galileo gives the times of rotation as approximately, (i.) 1 d. 18-1/2 h.; (ii.) 3 d. 13-1/3 h.; (iii.) 7 d. 4 h.; (iv.) 16 d. 18 h.; he also published configurations of the satellites calculated for March, April, and a part of May 1613. The periodic times of the satellites, as corrected by later observers, are, (i.) 1 d. 18 h. 28 m.; (ii.) 3 d. 13 h. 15 m.; (iii.) 7 d. 3 h. 43 m.; (iv.) 16 d. 16 h. 32 m.

[18] Modern astronomers agree in assigning an atmosphere to Jupiter, but consider it not extensive enough to affect the brightness of the satellites.—(Webb, Celestial Objects for Common Telescopes.) Their absolute magnitudes are different, and their surfaces have been observed to be obscured by spots, which may account for the variations of their brightness. These spots, like the lunar spots, are probably due to variations of reflective power at different parts of their surfaces, for as they always turn the same face to Jupiter, they present different portions of their surfaces to us periodically, and it has been ascertained by observation that “these fluctuations in their brightness are periodical, depending on their position with respect to the Sun.”—(Herschel, Outlines of Astronomy; Arago, Astronomie Populaire, 1854.)

[19] Diodorus Siculus, ii. 47.

[20] Kepler says in his introduction to his Commentaries upon the Motions of the Planet Mars, that the theory of gravitation depends on certain axioms, one of which is that “heavy bodies do not tend to the centre of the universe, supposing the earth to be placed there, because that point is the centre of the universe, but because it is the centre of the earth. So, wherever the earth be set, or whithersoever it be transported, heavy bodies have a continual tendency to it.” Kepler’s object in this work was to correct the methods for determining the apparent places of the planets according to the three theories then current—the Ptolemaic, the Copernican, and that of Tycho Brahe.

In 1593 the observed place of the planet Mars differed by nearly 5° from the place calculated for it. Kepler accordingly studied the motions of this planet, and “by most laborious demonstrations and discussions of many observations,” arrived at the conclusions known as Kepler’s first and second laws; according to which the Copernican system of eccentrics and epicycles was replaced by an ellipse whose centre and eccentricity were the same as the centre and eccentricity of the eccentric in the older method, and the Sun therefore was in one of the foci. Also the motion of the planet in its orbit was such that equal areas were described about the Sun by the radius vector of the planet in equal times.—Kepler, Astronomia Nova a?t??????t?? (Prague), 1609.

[21] The degree of accuracy attained by Kepler and Galileo with their imperfect instruments will be appreciated by comparing these statements with the determinations of later astronomers. Jupiter is about 1300 times the size of the Earth. Its diameter is about 87,000 miles; time of rotation, 9 h. 55 m. 21 sec.; time of revolution, 4333 days nearly. The angular diameter of the sun, seen from Jupiter, is between 6´ and 7´. The times of revolution of the four satellites are, as already given: (i.) 1 d. 18 h. 28 m., (ii.) 3 d. 13 h. 15 m., (iii.) 7 d. 3 h. 43 m., (iv.) 16 d. 16 h. 32 m.

[22] Umbistineum. Apparently this is some German word with a Latin ending, such as um-bei-stehn; Kepler fancied that Galileo had discovered two satellites of Mars.

[23] The text of the four letters of Galileo followed here is that given in the edition of Galileo’s works published at Florence, 1842-56; that in the edition of Kepler’s Dioptrics, published at Augsburg, 1611, is very inaccurate. These letters were written to Giuliano de’ Medici, ambassador of the Grand-Duke of Tuscany to the Emperor Rudolf II. at Prague.

[24] Virgil, Eclog. iii. 105.

[25] The completion of Galileo’s observations on Saturn depended on the improvement of astronomical instruments, as will be evident from the following sketch. Galileo made out the first indications of Saturn’s ring in 1610, as narrated in his letter, with a power of thirty; but in December 1612 he wrote to one of his friends, Marco Velseri, that he could no longer see these indications, and began to imagine that his telescope had deceived him, and apparently abandoned further researches. Hevelius in 1642 saw the ring more clearly, but figured it as two crescents attached to Saturn by their cusps. At length, in 1653, Huyghens provided himself with a power of one hundred, having made the lenses with his own hands, and immediately discovered the explanation of the phenomena which had baffled previous observers. He published his explanation of Saturn’s ring, and his discovery of the first satellite, in his Systema Saturnium, 1659. Cassini, with still more powerful instruments, discovered four more satellites in 1671, 1672, 1684. Sir William Herschel in 1789 detected two more, “which can only be seen with telescopes of extraordinary power and perfection, and under the most favourable atmospheric circumstances.”—(Herschel, Outlines of Astronomy, § 548.) And the last of the eight satellites was discovered in 1848 by Lassell of Liverpool, and Bond of Cambridge, U.S., simultaneously.

[26] Kepler, in his Mystery of the Universe, endeavoured to connect the orbits of the planets with the five regular solids, thus: If in a sphere (i.) a cube be inscribed, and in the cube a sphere (ii.); and in that sphere a tetrahedron, and in the tetrahedron a sphere (iii.); and in that sphere a dodecahedron, and in the dodecahedron a sphere (iv.); and in that sphere an icosahedron, and in the icosahedron a sphere (v.); and in that sphere an octahedron, and in the octahedron a sphere (vi.), the diameters of these six spheres will be proportional to the diameters of the orbits of Saturn, Jupiter, Mars, the Earth, Venus, and Mercury respectively; or, as Kepler expresses it, the common centre of these spheres represents the position of the Sun, and the six spheres represent the spheres of the planets.

By these considerations, however, Kepler was led to enunciate his third law, that the squares of the periodic times of planets are proportional to the cubes of their mean distances from the sun.—Kepler, Prodromus Dissertationum Mathematicarum continens Mysterium Cosmographicum, etc. (TÜbingen, 1596.)

[27] In the Ptolemaic system the earth’s centre was regarded as the centre of the universe, and the movements of the heavenly bodies were explained by eccentrics and epicycles. The sun was conceived to describe a circle about a point not exactly coinciding with the centre of the earth, called the sun’s eccentric. The planets described epicycles (circles) whose centres described eccentrics (circles), and the centres of these eccentrics coincided with the centre of the sun’s eccentric. In the case of Mercury and Venus the centre of the epicycle was always on the line drawn from the centre of the eccentric to the sun’s centre. In the case of the other planets the construction was more complicated. The stationary points were determined by drawing tangents from the earth’s centre (or the observer) to the epicycle, as in the figure (1).—(Gassendi, Institutio Astronomica, 1647.) This will explain Kepler’s description of the stationary points as the points where the planet leaves the tangent to its epicycle, supposing that he uses the terms of the current (i.e. Ptolemaic) astronomy. Copernicus placed the sun instead of the earth at the centre of the universe, but to determine the positions of the planets at any given time with as much accuracy as was attainable with the Ptolemaic system, he was obliged to use a similar method of eccentrics and epicycles, so that Kepler’s expression may be understood to describe the stationary points according to the Copernican theory, though it is still strange that he should not recognise the elliptical form of the planetary orbits, which he had lately demonstrated after most laborious reasoning in his Commentaries on the Motion of the Planet Mars, 1609. Galileo’s own expression seems to describe the stationary points according to the Copernican system, as would be expected, as the points where the planet leaves the tangent drawn to its orbit from the earth (Fig. 2).

[28] Lucian, Ver. Hist. i. 12.

[29] The first scientific determination of the period of the rotation of Venus was made by Dominique Cassini in 1666, from observations of spots on the planet, and concluded to be about 24 hours; but in 1726 Bianchini deduced a period of 24 d. 8 h. from similar observations. The true period is considered to be 23 h. 21 m., determined by Schroeter by a series of observations lasting from 1788 to 1793 on the periodicity of the deformation of the horns of Venus.—(Arago, Astronomie Populaire, 1854.)

Kepler’s statements can only be regarded as anticipations of phenomena not yet actually observed.

[30] Proctor (Other Worlds than Ours, 1875) has given some reasons for believing that Jupiter and Saturn shine in part with their own light, owing to their great internal heat.